This document contains 11 questions regarding differential equations. It asks the student to classify equations as order, linear or non-linear; formulate differential equations by eliminating constants; solve specific differential equations; prove properties of curves; find general solutions by substitution; and determine conditions for exactness and solve exact differential equations.

1. MA 108——Tutorial Sheet No. 1
Q.1. Classify the following equations (order, linear or non-linear):
(i) d3y
dx3 + 4(dy
dx)2 = y (ii) dy
dx + 2y = sin x (iii) y d2y
dx2 + 2xdy
dx + y = 0
(iv) d4y
dx4 + (sin x)dy
dx + x2y = 0. (v)(1 + y2)d2y
dt2 + td6y
dt6 + y = et.
Q.2. Formulate the differential equations represented by the following functions by eliminating the arbi-
trary constants a, b and c:
(i) y = ax2 (ii) y − a2 = a(x − b)2 (iii) x2 + y2 = a2 (iv) (x − a)2 + (y − b)2 = a2
(v) y = a sin x + b cos x + a (vi) y = a(1 − x2) + bx + cx3 (vii) y = cx + f(c).
Also state the order of the equations obtained.
Q.3. Solve the equation x3(sin y)y = 2. Find the particular solution such that y(x) → π
2 as x → +∞.
Q.4. Prove that a curve with the property that all its normals pass through a point is a circle.
Q.5. Let ϕi be a solution of y + ay = bi(x) for i = 1, 2.
Show that ϕ1 + ϕ2 satisfies y + ay = b1(x) + b2(x). Use this result to find the solutions of
y + y = sin x + 3 cos 2x passing through the origin.
Q.6. Obtain the solution of the following differential equations:
(i) (x2 + 1)dy + (y2 + 4)dx = 0; y(1) = 0 (ii) y = y cot x; y(π/2) = 1
(iii) y = y(y2 − 1), with y(0) = 2 or y(0) = 1, or y(0) = 0 (iv) (x + 2)y − xy = 0; y(0) = 1
(v) y +
y − x
y + x
= 0; y(1) = 1 (vi) y = (y − x)2; y(0) = 2
(vii) 2(y sin 2x + cos 2x)dx = cos 2xdy; y(π) = 0. (viii) y =
1
(x + 1)(x2 + 1)
Q.7. For each of the following differential equations, find the general solution (by substituting y = vx)
(i) y =
y2 − xy
x2 + xy
(ii) x2
y = y2
+ xy + x2
(iii) xy = y + x cos2
(y/x) (iv) xy = y(ln y − ln x)
Q.8. Show that the differential equation
dy
dx
=
ax + by + m
cx + dy + n
where a, b, c, d, m and n are constants can be
reduced to
dy
dt
=
ax + by
cx + dy
if ad − bc = 0. Then find the general solution of
(i) (1 + x − 2y) + y (4x − 3y − 6) = 0
(ii) y = y−x+1
y−x+5
(iii) (x + 2y + 3) + (2x + 4y − 1)y = 0.
Q.9. Solve the differential equation 1 − y2dx+
√
1 − x2dy = 0 with the conditions y(0) =
±1
2
√
3. Sketch
the graphs of the solutions and show that they are each arcs of the same ellipse. Also show that after
these arcs are removed, the remaining part of the ellipse does not satisfy the differential equation.
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2. Q.10. State the conditions under which the following equations are exact.
(i) [f(x) + g(y)]dx + [h(x) + k(y)]dy = 0
(ii) (x3 + xy2)dx + (ax2y + bxy2)dy = 0
(iii) (ax2 + 2bxy + cy2)dx + (bx2 + 2cxy + gy2)dy = 0
Q.11. Solve the following exact equations
(i) 3x(xy − 2)dx + (x3 + 2y)dy = 0 (ii) (cos x cos y − cot x)dx − sin x sin ydy = 0.
(iii) exy(x + y)dx + ex(x + 2y − 1)dy = 0
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